What does it mean to quantize a field

[copernicus1]

Textbooks I've looked into usually follow a sort of "prescription" for quantizing a field theory, which consists of "promoting the fields to operators" and "imposing commutation relations." Is there an English translation of what this means? Like for instance, is this tantamount to taking the classical continuous field and endowing it with a "quantum"---that is, a discrete particle description? Is that the main difference between the classical and quantized fields?

 

[psmt]

Like for instance, is this tantamount to taking the classical continuous field and endowing it with a "quantum"---that is, a discrete particle description?

Yes - bear in mind that like any quantum theory, a quantum field theory has a state vector, although because the number of particles is not fixed, this state vector belongs in a Fock space rather than a Hilbert space. For a non-interacting theory, the particle number operator commutes with the Hamiltonian, so energy eigenstates are particle-number eigenstates. The vacuum state (i.e. lowest-energy eigenstate of the Hamiltonian) is also a state with no particles. States with a definite number of particles are produced by acting on the vacuum state with creation operators, which are derived from the field operators.

Is this helpful, or am i just repeating what the textbooks are saying?

 

[Sonderval]

@copernicus
To get an intuition of what "quantizing a field" should mean, think about what "quantizing a particle" actually means: It means (among other things) that you cannot say that the particle is at a position x, but rather that there is a probability amplitude for it to be at a position x.
Transferring this to a field, you see that a classical field has a certain value of the field at each point in spacetime. Quantizing this then implies that instead you have a probability amplitude for each field configuration.

The trouble is that this means to define a kind of wave function that takes a function as an argument (since each field configuration is in itself a function). That's pretty awkward mathematically (although it can be done).

This it is more convenient to copy the knowldege from QM that you can rewrite all the wave function stuff using operators, like Heisenberg did. If you impose the commutation rule for space and momentum for a particle, this brings you to the same conclusions as the wave function picture. So you then transfer this method to the fields and impose the commutation rules on the field.

As further reading, I would recommend (although some people hate it due to some idiosyncrasies) to read chapter 3 of bob klaubers intro to QFT.

 

[copernicus1]

Thanks Sonderval, this is pretty helpful. I actually like Klauber's QFT site.

 

[phys_student1]

@copernicus
As further reading, I would recommend (although some people hate it due to some idiosyncrasies) to read chapter 3 of bob klaubers intro to QFT.

I do not see any problem with this, since the author CLEARLY states what is his own personal take (and often tells you that the "standard" is different".

Nonetheless, this only happens in rare places, like vacuum expectation etc. I am reading the book now and can only recommend it (I have been frustrated by Peskin & Schroeder and others).

Again, I highly recommend Klauber's text.